Whether or not you use a car service is up to you. I've never priced the car services in the metro area since we're too far away for that to be an option. Given you aren't leaving until late in the day on Thanksgiving AND that this holiday is heavily traveled, it might be well worth it to hire a car to drive you to the airport. I'm sure the longterm parking (especially economy lots) will be crammed.
In the event you decide to drive yourselves and are able to get a spot (listen to WJR--they give those lot reports all the time OR call the airport to check parking availability prior to leaving), the cost of parking is higher at the airport than say Park & Fly or Park & Jet or any of the other parking services just across 94. I think in the past we've paid between $60 and $90 depending on the length of our trips (the Davey parking structure is higher than the economy lots and I've not flown since McNamara has opened). The private lots are a bit less expensive and equally secure. We've never had a vehicle tampered with at either the airport or the private lots. We've also had great luck catching the transportation vehicles to/from the lots when we arrive and return.
As to your question about whether or not metro will be "a madhouse"--you know it will! I would say that you're probably better off with a late day flight than the early a.m. BUT you might encounter flight delays due to increased traffic.
Good luck!